PREDICTING THE PROBABILITY OF FAILURE OF GAS PIPELINES INCLUDING INSPECTION AND REPAIR PROCEDURES

Size: px

Start display at page:

Download "PREDICTING THE PROBABILITY OF FAILURE OF GAS PIPELINES INCLUDING INSPECTION AND REPAIR PROCEDURES"

Darrell White
5 years ago
Views:

1 REDICTING THE ROBABILITY OF FAILURE OF GAS IELINES INCLUDING INSECTION AND REAIR ROCEDURES Zhang, L. 1, Adey, R.A. 2 1 C M BEASY Ltd, Ashurst Lodge, Lyndhurst Road, Southampton, SO40 7AA, UK, lzhang@beasy.com 2 C M BEASY Ltd, Ashurst Lodge, Lyndhurst Road, Southampton, SO40 7AA, UK, r.adey@beasy.com ABSTRACT This paper is concerned with predicting the impact on the probability of failure of adding hydrogen to the natural gas distribution network. Hydrogen has been demonstrated to change the behaviour of crack like defects which may affect the safety of pipeline or make it more expensive to operate. A tool has been developed based on a stochastic approach to assess the failure probability of the gas pipeline due to the existence of crack-lie defects including the operational aspects of the pipeline such as inspection and repair procedures. With various parameters such as crack sizes, material properties, internal pressure modelled as uncertainties, a reliability analysis based on failure assessment diagram is performed through direct Monte Carlo simulation. Inspection and repair procedures are included in the simulation to enable realistic pipeline maintenance scenarios to be simulated. In the data preparation process, the accuracy of the probabilistic definition of the uncertainties is crucial as the results are very sensitive to certain variables such as the crack depth, length and crack growth rate. The failure probabilities of each defect and the whole pipeline system can be obtained during simulation. Different inspection and repair criteria are available in the Monte Carlo simulation whereby an optimal maintenance strategy can be obtained by comparing different combinations of inspection and repair procedures. The simulation provides not only data on the probability of failure but also the predicted number of repairs required over the pipeline life thus providing data suitable for economic models of the pipeline management. This tool can be also used to satisfy certain target reliability requirement. An example is presented comparing a natural gas pipeline with a pipeline containing hydrogen. Nomenclature f xn (x N ) distribution function of variables c half crack length a crack depth t wall thickness g( x ) failure domain Φ standard normal distribution function x i stochastically independent variable K r ratio of applied elastic K to K IC L r ratio of applied load to yield load ΔK th threshold stress intensity factor value ρ plastic correction factor L r max permitted limit of L r σ Y yield strength of material σ U ultimate tensile strength of material K IC Toughness of material σ ref reference stress N number of cycles D / a probability of detection p f probability of failure of a single defect q total number of cracks C t total cost of repair program C s cost of repairing a single crack N A average number of cracks repaired C pf probable cost of have a failure f total total probability of failure C f cost of having a major failure β reliability index 1

2 1. INTRODUCTION Failure of a structural member such as a pipeline may occur when a crack propagates in an unstable manner to cause leak or explosion of the pipeline. Fracture mechanics combined with a probabilistic approach has been utilized in many fields of analysis involving important structural components such as pressure vessels and nuclear piping. Based on probabilistic fracture mechanics, statistical methods are applied in order to assess the reliability of pipeline containing crack-like defects [1], in other words, to provide a single number which represents the probability that a pipeline could fail. The aim of the NATURALHY project is to investigate the possibility of using the existing natural gas pipelines to deliver hydrogen or mixed natural-hydrogen gas. As zero tolerance to hydrogen leakage is widely accepted in the pipeline industry the concept of 'failure' of a pipeline includes both gas leakage and pipeline breakage. Although in general the safety of pipelines is threatened by corrosion and crack like defects this paper concentrates on crack like defects as data suggests that only crack like defects are affected by the presence of hydrogen. Under the slow fatigue cycles hydrogen can affect the fracture toughness and the rate of growth of cracks. 2.0 THEORY 2.1 Fundamentals of reliability analysis According to robabilistic Fracture Mechanics (FM) if one and only one crack exists in a pipeline, the failure probability is given by: f = f X (x 1 ) f X ( x N )d ( x 1 ) d ( x N ) (1) g ( x 1 x N ) 0 1 N where x 1 x N are random variables such as crack sizes, yield strength, crack growth parameters and applied stresses. As will be discussed in chapter 3, only semi-elliptical surface cracks will be considered as crack models with the crack depth a and crack length 2c. The geometry of the model is shown in Fig. 1. f x N (x N ) denotes the probability density function of the input variable x N. The integration will be performed over the failure domain where g(x 1 x N ) 0. Figure 1. Geometry of the Semi-elliptical surface crack model used For engineering applications a reliability index β is required to define the amount of safety of the structure, which is given by [2]: 2

3 f = Φ ( β ) (2) where Φ is the standard cumulative normal distribution function. 2.2 Failure assessment procedure In the current reliability analysis a Failure Assessment Diagram (FAD) approach is adopted, which is shown in Fig 2. The g-function aforementioned is based on the BS7910 level 2 Failure Assessment Curve (FAC). Figure 2. The FAC curve used in the calculation The FAC is defined as follows: for L r > Lr max K r = 0 for L L K = ( L 2 ){ exp( 0.65 L 6 r r max r r r )} (3) The failure criterion includes both brittle fracture and plastic collapse. K r measures the proximity to brittle fracture and L r represents the likelihood of plastic collapse. For BS7910 level 2A FAD, they are given by: K + K S K r = + ρ K IC σ ref L r = σ Y (4) ρ is a parameter that takes plastic interaction between primary and secondary stress into consideration. For materials that exhibit a yield discontinuity (referred to as Lüders plateau) L r is restricted to 1.0 [3]. Otherwise it is calculated through: 3

4 σ Y +σ u L r max = 2σ (5) Since the cumulative probability of failure over a given timeframe is required, crack propagation due to cyclic loading must be included. The ARIS law with a threshold ΔK th is selected to calculate the crack length and depth with regard to the corresponding number of cycles. da 0 for ΔK <ΔKth = dn CΔK m for ΔK Δ K th (6) The actual calculation in ipesafety is performed by estimating the amount of crack growth during a loading cycle. Δa = C(ΔK a ) m a = a + Δ a n+1 n (7) where a n corresponds to the crack depth after n load cycles. An equivalent equation applies for the second axis of the semi-ellipse. The initial crack depth and length are modelled as variables and if necessary the fatigue property parameters can be also modelled as random variables based on certain distributions. As crack propagation could lead to fracture or leakage of the pipeline after a certain period of time, f is a function of load cycle n. f = ( n ) (8) f denotes the cumulative probability which monotonically increases with time or load cycles. The inclusion of inspection and repair program can slow down this process so as to meet certain target reliability targets. 2.3 Inclusion of inspection and repair program As part of the normal operation and maintenance of pipelines, inspections are performed using intelligent pigs to detect defects in the pipeline. During inspections not all defects are identified due to the sensitivity of the tool. The process of inspection and repair of a pipeline at a given interval will change the distribution of crack depth and length because some of the detected cracks will be repaired. The exact distribution will depend upon the repair strategy adopted, the frequency of inspection and the sensitivity of the inspection tool. These remaining cracks will not lead to failure but those missed by the inspection tool might cause gas leakage or rupture of the pipeline. The maintenance event tree is displayed in Fig. 3. 4

5 Figure 3. Maintenance event tree (adapted from [4]) So far the robability of Detection (OD) has not been defined. The OD is usually an increasing function of defect depth and defined as an exponential function [4]: D / a = 1 e λa (9) D / a can be expressed as the cumulative distribution function for the detectable depth of the inspection tool. Hence, the detectable depth of the inspection tool follows the exponential distribution function and both the average detectable size and the standard deviation equal 1/ a. If inspection and repair program is introduced, the calculation of f is slightly more complex and the calculation is based on the intervals of the inspection and repair program [4]. If the pipeline is assumed to be running safety at n = 0 the cumulative OF before the first inspection is obtained as: f ( n ) = [ S (0) I F(N)]/ [ S (0)] (0 < n < N 1 ) (10) and the OF between the first inspection and the second inspection is given by: f ( n) = f (N) + f 1 (N) + f 2(N) ( N 1 < n < N 2 ) (11) f 1 ( n )= [ S (0) I S( N 1) I ND ( N 1) I F(N 1 )]/ [ S (0)] (12) f 2 ( n )= [ S (0) I S( N 1 ) I D( N 1 ) I NR ( N 1 ) I F(N 1)]/ [ S (0)] (13) Where S( n ) is an event representing a crack that survives between 0 and n cycles. Similarly, ND ( n ) refers to non-detection event; D( n ) means detection event; NR ( n ) is non-repair event and F( n ) refers to an event where a crack lies in the un-safe area. As can be seen from the equations above, when the cycle number is between 0 and N 1 the calculation of OF is straightforward. However, after the first inspection and repair program, the calculation of OF is composed of two parts. The first component f 1 ( n ) corresponds to the cracks that are un-detected and will lead to failure. f 2 ( n ) corresponds to the detected but un-repaired cracks that eventually result in failure. If more than one inspection program exists, similar expressions can be deduced. These expressions are 5

6 mathematical representation of the maintenance event tree that has been shown in Fig. 3. In addition to the probability of failure, the probability of repairing a single crack is obtained as follows: ( R n = N 1 )= [ S (0) I S( N 1 ) I D( N 1 ) I R( N 1 )]/ [ S (0)] (14) This expression means the repaired cracks are those that survived the first inspection and are detected and repaired by the inspection tool. There are three options for repair criteria: a) any cracks detected are repaired; b) only the cracks that are larger than a certain size are repaired; c) the cracks that will lead to failure after a pre-determined period of time will be repaired. This time period is usually defined by the pipeline operator. 3.0 MONTE CARLO SIMULATION 3.1 Monte Carlo simulation procedure Solution to Eq. (1) can be solved by Monte Carlo simulation. By generating a large number M of independent repetitions, the probability of failure can therefore be estimated as the quotient of the failure counts to the number of simulations performed, which is given as follows: f = M f M (15) The above equation is valid when the pipeline only contains one crack. When there is more than one crack in the system, say q stochastically independent cracks, the total probability of failure is the probability that at least one crack will lead to failure, which is obtained by [2,4]: f total = 1 (1 f ) q (16) The scheme of the Monte Carlo simulation is presented in Fig. 4. The first step is to generate random numbers based on the selected probability distribution functions. These variables include the crack depth, length, material properties, geometry of the pipe, etc. The simulation follows the same logic as is illustrated in the previous chapters. 3.2 Random variables and sampling method The defect sizes given by non-destructive testing can follow a normal, lognormal or exponential distribution [5]. In addition, the distributions of other parameters have been investigated by some authors [6-8]. Once a random number between 0 and 1 has been generated, it can be used to generate required random variables with a given probability distribution function. The inverse transform method [9] is a common method to achieve this, which is based on the observation that continuous cumulative distribution functions (cdf) range uniformly over the interval (0, 1). If u is a uniform random number on (0, 1), then a random number x from a continuous distribution with selected cdf F can be obtained using: x = F 1 x ( u ) (16) 6

7 However, for normal distribution the inverse of the cumulative distribution function cannot be found analytically. Hence, other methods such as the Box-Müller method or the envelop-rejection method is used in the simulation [9]. There are two sampling methods for Monte Carlo simulation: a) Direct Monte Carlo simulation b) Monte Carlo simulation with variance reduction: stratified sampling or importance sampling The software developed for the current project employs both direct Monte-Carlo method and Monte- Carlo method with stratified sampling. In this paper the direct Monte Carlo simulation has been explained and the detailed information about the variance reduction techniques can be found in literature [8]. No End analysis Generate random variables? Yes No Detected? Yes No Repair? Yes No failure Grow the crack for inspection interval time t Fail? Yes Mf=Mf+1 No Yes More inspections? No No failure Figure 4. Flow chart of OF Monte Carlo analysis 7

8 4. DEVELOMENT OF FM TOOL-IESAFETY Fig. 5 shows the first page of the OF calculation software ipesafety. The software is composed of six modules, which includes geometry definition, load definition, material property definition, inspection tool definition, repair criteria definition and finally the settings of calculation. 5.0 SAMLE ROBLEMS 5.1 Data preparation Figure 5. Main page of ipesafety software The following examples will illustrate the OF calculation and maintenance procedure of an X52 pipeline based on the input data listed in Table 1. The aim is to investigate the impact of the introduction of hydrogen on the total probability of failure within a given timeframe. We assume the pipeline is pre-inspected and the total number of cracks per km in base material and weld is 10cracks/km respectively [11]. Also all cracks are assumed to be longitudinally oriented. Table 1. Input parameters for OF analysis of X52 steel in natural gas and hydrogen pipeline No. arameter Average Standard deviation Distribution type 1 ipe diameter (mm) Fixed 2 Wall thickness (mm) 10 0 Fixed 3 Initial crack depth (mm) Log-normal 8

9 No. arameter Average Standard deviation Distribution type 4 Initial crack length (mm) Log-normal 5 ressure (Ma) Fixed 6 Residual stress(ma) Base metal or weld with 0 Fixed 7 Fracture toughness (Ma*mm 1/2 ) relaxation Assumed for N 2 and for H 2 [10] 0 Fixed 8 Yield strength (Ma) Normal 9 Tensile strength (Ma) Normal 10 Threshold toughness (Ma*mm 1/2 ) Fixed 11 Fatigue parameter c fixed (Natural gas) [10] 12 Fatigue parameter m (Natural gas) 3[10] 0 fixed 13 Fatigue parameter c [10] 0 fixed (Hydrogen) 14 Fatigue parameter m 3.8[10] 0 fixed (Hydrogen) 15 Inspection interval re-inspected and will be inspected at every 10 years 0 fixed 15 Inspection tool precision 95% probability of detection 4% wall thickness 16 Repair criterion Repaired if the crack will lead to failure in 12 years 0 fixed 0 fixed 17 ressure drop ratio fixed 18 Service life (years) 40 0 fixed 19 No. of cycles per year fixed 5.2 robabilistic fracture failure analysis results Using ipesafety software, the cumulative probability of failure can be obtained as shown in Table 2. As can been seen from the results, introduction of 100% hydrogen will significantly affect the reliability of the pipeline and therefore extra care has to been taken when dealing with hydrogen pipelines. The average number of cracks repaired is also computed and the total cost of inspection and repair program can be calculated by: C = N C t A s (16) where C is total cost; N is the average number of cracks repaired and C is the cost of repairing a t A s single defect. Similarly the probable cost of failure is: 9

10 C = C pf f total f (17) where C pf is the probable cost of failure and f total is the total probability of failure and C f is the cost of a major failure. Table 2. Monte Carlo simulation results of the X52 pipeline in different gas media Cracks in natural gas 5 ( t = 40 years) 3 10 f Cracks in Hydrogen Cracks in weld and natural gas Cracks in weld and hydrogen Average number of cracks repaired f total Figure 6. OF vs. standard deviation ("divided by 100" means OF has been scaled down by 100) It is well known that the OF is heavily dependent on the distribution of the defects in the pipeline and in particular the crack depth [12]. Fig. 6 shows how the OF changes as the standard deviation of crack depth increases. Fortunately many pipelines have been operating safely carrying Natural gas for many years which enables the Standard Deviation to be estimated from current safety data. Therefore in Fig. 6 it is shown that if the current OF of a gas pipeline is 10-5 then for the same pipeline carrying 100% hydrogen the OF will increase to 10-4 assuming it is operated in the same manner. If pipelines carrying natural gas-hydrogen mixture are used we expect the OF could be lower than that of the pure hydrogen pipelines. However the fatigue properties of X52 in natural gas-hydrogen mixture are not clear now and massive experiments are being performed. Once sufficient data are obtained OF of different types of pipeline can be predicted using the same approach as proposed. 10

11 6. CONCLUSIONS A methodology based on BS7910 has been presented for assessment of the safety of pipelines carrying hydrogen or natural gas. The methodology is based on Monte-Carlo simulation which includes the assessment of crack like defects and the inspection and repair strategy. The existence of hydrogen in a pipeline significantly affects the safety of the pipeline and the probability of failure for all the cases considered. The reason for this is the change in the fatigue and fracture properties of the material induced by hydrogen. Another important element of the OF calculation is the assumed distribution of crack like defects present in the pipeline at the start of its operation with hydrogen. For example a new pipeline will have a smaller number of initial defects than an older one. As can be seen from the results of the examples, different input data such as mean value or standard deviation of the crack sizes can lead to very different OF results. Other service conditions such has pressure drop ratio, number of cycles and the type of defects (e.g. base metal defects, weld defects, crack like defects associated with third party damage) have to be considered. Hence, when calculating OF much importance should be attached to the way how this data is collected and used. There are several assumptions made in this paper that need to be noted: a) Crack size distribution In this report, only lognormal function is adopted to represent the actual distribution of the defects in pipeline. However, there are other functions such as Weibull and exponential distributions, which can also be used to fit the data. In addition, the crack length and depth should be examined very carefully since the results could become rather different for any slight change of the input data. b) Number of cracks in welds Currently it is impossible to know the exact number of cracks in the pipeline so sensible assumptions have to be made. In this report, the assumed number of defects is based on TNO's report [11]. c) Detection probability Detection probability is essentially related to the distribution of the crack depth. The larger the crack depth is, the higher possibility that a crack can be detected by the inspection tool is. In the current analysis it was assumed the minimum detectable depth is 0 mm but this might not reflect the real precision of the inspection tool. A shifted OD curve is used for more realistic situation where the minimum detectable depth is larger than 0 as shown on the right hand side of Fig. 7. Figure 7. OD vs. crack depth 11

12 This paper presents a probabilistic approach to estimate the pipeline reliability incorporating inspection and repair program over the service life of the pipeline which contains crack-like defects. Monte Carlo simulation based software has been developed to calculate the OF and help engineers and decision makers of the pipeline operators to determine the optimal inspection and repair interval. The data presented show that the introduction of 100% hydrogen to an existing natural gas pipeline with crack-like defects will dramatically increase the OF of the pipeline system if the pipeline is continued to be inspected and repaired in the same way. The current work is extending the calculations to natural gas and hydrogen mixtures. ACKNOWLEDGEMENTS The authors would like to thank the NATURALHY project partners for providing data and information and the European Union for kindly providing the financial support for this study. REFERENCES 1. A.Bruckner and D.Munz, Determination of Crack Size Distributions from Incomplete Data Sets for the Calculation of Failure robabilities. Reliability Engineering, : p Thoft-Christensen,. and M.J. Baker, Structural Reliability Theory and Its Applications. 1982: Springer-Verlag. 3. Guide to methods for assessing the acceptability of flaws in metallic structures (BS7910). 2005: British Standards Institution. 4. M.D.andey, robabilistic Models for Condition Assessment of Oil and Gas ipelines. NDT&E International, (5): p Y.C.Lin and Y.J.Xie, Expert System for Integrity Assessment of iping Containing Defects. Expert Systems with Applications, : p G.Walz and H.Riesch-Oppermann, robabilistic Fracture Mechanics Assessment of Flaws in Turbine Disks Including Quality Assurance rocedures. Structural Safety, : p H.Adib, et al., Evaluation of the effect of corrosion defects on the structural integrity of X52 gas pipelines using the SINTA procedure and notch theory. ressure Vessels and iping, : p W.rovan, J., ed. robabilistic Fracture Mechanics and Reliability. Engineering Application of Fracture Mechanics, ed. G.C. Sih. Vol , Martinus Nijhoff ublishers. 9. J.S.Dagpunar, Simulation and Monte Carlo. 2007: John Wiley & Sons Ltd. 10. Krom, A., L. Zhang, and R. Adey, W4 TASK B: Critical defect sensitivity studies. 2006, TNO Netherlands & CMI UK. 11. Wortel, H.v., robability of Detection (od) and robability of Failure (of) of flaws in the natural gas network (first draft). 2006, TNO. 12. A.Bruckner and D.Munz, The Effect of Curve Fitting on the rediction of Failure robabilities from the Scatter in Crack Geometry and Fracture Toughness. Reliability Engineering, : p

ASSESSMENT OF THE PROBABILITY OF FAILURE OF REACTOR VESSELS AFTER WARM PRE-STRESSING USING MONTE CARLO SIMILATIONS

Int J Fract DOI 10.1007/s10704-012-9800-5 Springer Science+Business Media Dordrecht 2012 LETTERS IN FRACTURE AND MICROMECHANICS ASSESSMENT OF THE PROBABILITY OF FAILURE OF REACTOR VESSELS AFTER WARM PRE-STRESSING